Abstract:The Chern-Simons theory is a gauge theory which is a version of (2+1)-dimensional TQFT (topological quantum field theory). It provided a useful framework and tools to understand the topology of knots in a 3-manifold, for example, the Jones polynomial of knots. The arithmetic Chern-Simons theory, initiated by Minhyong Kim, is an arithmetic analogue of the Chern-Simons theory, which is expected to attack the number theory problem (Galois theory problem, L-functions, Iwasawa theory, and etc) guided by physics (quantum field theory) and topology principles and techniques appearing in the Chern-Simons theory. In this talk, we will briefly explain the analogy between primes in a number field and knots in a 3-manifold, and define the arithmetic Chern-Simons action. Then we will provide arithmetic applications to a certain Galois embedding problem based on an explicit computation of the arithmetic Chern-Simons action. This is a joint work with Hee-Joong Chung, Dohyeong Kim, Minhyong Kim, and Hwajong Yoo.